Properties

Label 2-5054-1.1-c1-0-7
Degree $2$
Conductor $5054$
Sign $1$
Analytic cond. $40.3563$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.70·3-s + 4-s − 3.24·5-s − 1.70·6-s − 7-s + 8-s − 0.0783·9-s − 3.24·10-s − 4.04·11-s − 1.70·12-s + 3.87·13-s − 14-s + 5.55·15-s + 16-s − 6.97·17-s − 0.0783·18-s − 3.24·20-s + 1.70·21-s − 4.04·22-s − 4.26·23-s − 1.70·24-s + 5.55·25-s + 3.87·26-s + 5.26·27-s − 28-s − 6.78·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.986·3-s + 0.5·4-s − 1.45·5-s − 0.697·6-s − 0.377·7-s + 0.353·8-s − 0.0261·9-s − 1.02·10-s − 1.22·11-s − 0.493·12-s + 1.07·13-s − 0.267·14-s + 1.43·15-s + 0.250·16-s − 1.69·17-s − 0.0184·18-s − 0.726·20-s + 0.372·21-s − 0.863·22-s − 0.888·23-s − 0.348·24-s + 1.11·25-s + 0.760·26-s + 1.01·27-s − 0.188·28-s − 1.26·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(40.3563\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4184631950\)
\(L(\frac12)\) \(\approx\) \(0.4184631950\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
19 \( 1 \)
good3 \( 1 + 1.70T + 3T^{2} \)
5 \( 1 + 3.24T + 5T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 - 3.87T + 13T^{2} \)
17 \( 1 + 6.97T + 17T^{2} \)
23 \( 1 + 4.26T + 23T^{2} \)
29 \( 1 + 6.78T + 29T^{2} \)
31 \( 1 - 2.78T + 31T^{2} \)
37 \( 1 + 6.34T + 37T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 + 8.78T + 43T^{2} \)
47 \( 1 + 9.31T + 47T^{2} \)
53 \( 1 + 5.89T + 53T^{2} \)
59 \( 1 + 0.510T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 - 4.02T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 9.94T + 79T^{2} \)
83 \( 1 + 3.24T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 6.78T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.302102883936681255996703854218, −7.32078046509571278137881156521, −6.63597641916640450145939870363, −6.07407115875079445374395740232, −5.19817208557314650395323631496, −4.64284239723708763038632214223, −3.77914392703761566422114610403, −3.19738940232253914371764237745, −2.00581902351187474212639035349, −0.31242819526500941749571406703, 0.31242819526500941749571406703, 2.00581902351187474212639035349, 3.19738940232253914371764237745, 3.77914392703761566422114610403, 4.64284239723708763038632214223, 5.19817208557314650395323631496, 6.07407115875079445374395740232, 6.63597641916640450145939870363, 7.32078046509571278137881156521, 8.302102883936681255996703854218

Graph of the $Z$-function along the critical line